Sharp lower and upper bounds for the covariance of bounded random variables

In this paper we derive sharp lower and upper bounds for the covariance of two bounded random variables which are applicable when knowledge about their expected values, variances or both is available. When only values known, our result can be viewed as an extension Bhatia–Davis inequality variances. We also provide a number different ways to standardize covariance. For binary pair variables, one these standardized measures covariation agrees with frequently used measure dependence between genetic variants.